The three primary functions of a drilling fluid—the transport of cuttings out of the wellbore, prevention of fluid influx, and the maintenance of wellbore stability—depend on the flow of drilling fluids and the pressures associated with that flow. For example, if the wellbore pressure exceeds the fracture pressure, fluids will be lost to the formation. If the wellbore pressure falls below the pore pressure, fluids will flow into the wellbore, perhaps causing a blowout. It is clear that accurate wellbore pressure prediction is necessary. To properly engineer a drilling fluid system, it is necessary to be able to predict pressures and flows of fluids in the wellbore. The purpose of this chapter is to describe in detail the calculations necessary to predict the flow performance of various drilling fluids for the variety of operations used in drilling and completing a well.

Overview

Drilling fluids range from relatively incompressible fluids, such as water and brines, to very compressible fluids, such as air and foam. Fluid mechanics problems range from the simplicity of a static fluid to the complexity of dynamic surge pressures associated with running pipe or casing into the hole. This chapter first presents a general overview of one-dimensional (1D) fluid flow so that the common features of all these problems can be studied. Next, each specific wellbore flow problem is examined in detail, starting from the simplest and progressing to the most complicated. These problems are considered in the following order:

Static incompressible fluids.

Static compressible fluids.

Circulation of incompressible fluids.

Circulation of compressible fluids.

General wellbore steady flow.

Steady-state surge pressure prediction.

Following these basic problems, a series of special topics are presented:

Governing Equations

A complete fluid mechanics analysis of wellbore flow solves the equations of mass, momentum, and energy for each flow stream and the energy equation for the wellbore and formation. In the usual treatment of these equations, the mass conservation equations are not stated explicitly, and the temperatures are given, rather than calculated from the energy equation. Here, the whole problem is set out, with appropriate assumptions made for special, simplified cases as they are considered.

The flow streams are treated as 1D constant area flow but with recognition of the effects of discontinuous area changes, such as nozzles. The assumption of 1D flow means that the flow variables, such as density, velocity, viscosity, etc., are given as their average values over the cross-sectional area of the flow stream. For instance, for flow in a tube, the frictional pressure drop is formulated in terms of the average velocity, density, and viscosity. The equations of mass and momentum conservation are solved subject to the assumption of steady flow. This assumption is that time variations of all variables are neglected in a time increment. In particular, this means that mass accumulation effects are not considered in the mass balance equation and that velocity is only a function of position in the momentum balance equation. Fully dynamic momentum equations are considered in a later section.

The balance equations are written in a control volume form. The equations are written as integrals over a specified volume with specified surface areas rather than as partial differential equations at a point. For these flow streams, the volume under consideration has cross-sectional area A and length Δz . The surface areas are the circular or annular cross-sectional areas, A, of the ends and the cylindrical lateral surface area. In the solution of these equations, only the entrance and exit values of the flow variables are calculated. To evaluate the integrals, the variation of variables between the entrance and exit of each space increment may be needed. The usual assumption used in wellbore calculations is that density, velocity, viscosity, and thermal conductivity are constant and equal to entrance conditions through the increment and that pressure and temperature vary linearly between entrance and exit. Experience has shown these assumptions to be good except for compressible flow. For most cases, the increased accuracy from other interpolation methods does not justify the computation penalty. On the other hand, none of these calculations are so numerically intensive that they cannot be done with more accurate integration methods on a personal computer, and some of these methods are mentioned later in the text.

Single-Phase Flow

where steady flow (time independent flow) has been assumed. By Eq. 3.1 , the mass flow rate in any flow stream is constant. In other words, the rate of fluid flow into a volume equals the rate of fluid flow out of the volume. Note that this relation does not change with area changes. However for nonsteady-state flow, we find that mass can accumulate in the volume so that flow out does not necessarily equal flow in.

where steady flow has again been assumed. The Δv term is called the fluid acceleration and is important only for compressible fluids. The ρg term is the fluid weight term, which is positive for flow downward. The ρv2 term is the fluid friction term. The Fanning friction factor f depends on the fluid density, velocity, viscosity, fluid type, and pipe roughness. Appropriate models for f, considering a variety of different fluid types, are considered in detail in the section on rheology. The sign of the friction term is counter to the flow direction (e.g., negative for flow in the positive direction). For area changes, the following relation holds

....................(3.3)

Eq. 3.3 simplifies for incompressible flow and is written as

....................(3.4)

where subscript o indicates upstream properties, and subscript 1 indicates downstream properties. The quantities Cd are the discharge coefficients for the flow through an area change. Exact treatment of the effect of area change on momentum would have Cd equal to 1. However, real flow is not 1D through area changes, so a factor is needed to account for the real three-dimensional (3D) flow effects. Flow into a smaller area results in a reversible pressure drop plus an irreversible pressure drop. Flow into a larger area results in a reversible pressure increase plus an irreversible pressure drop. Thus, the values of Cd are different for flow into a restriction (reduced area) and flow out of a restriction (increased area). The following values of Cd are typical:

Cd = .95 into reduced area. Cd = 1.00 into increased area.

The basic balance of energy equation for single-phase flow is given by

This equation is given in the fully transient form because temperature variation with time may be significant and because steady-state temperatures are usually not achieved in typical wellbore hydraulic operations. The viscous dissipation term Φ depends on the fluid friction model. The term Q is usually written as the total heat flux into the control volume. An example of Θ would be the heat of hydration for cement. Eq. 3.3 can be rewritten in terms of enthalpy h.

....................(3.6)

and by choosing pressure and temperature as independent variables can be further rewritten as

Key Considerations for Wellbore Hydraulic Simulation

While many applications can be done as hand calculations, more complex problems, especially involving temperature changes, require a hydraulic simulator. To address the wellbore operations of interest, a wellbore simulator should have a wide range of capabilities. These fall into four categories:

1. Transient effects.

2. Fluid models.

3. Wellbore geometry.

4. Flow types.

Many applications for operational design involve highly transient behavior where temperatures are changing rapidly. Drilling, cementing, fracturing, and production startup are all transient operations where fluid temperatures can change on the order of 100°F or more in a matter of minutes during flow in the well. Fully transient thermal response should be modeled in the flowing stream, the wellbore assembly, and the formation. The model should handle changing flow conditions, including changes in flow rate, inlet temperature and pressure, fluid type, and flow direction.

Oil and gas well operations involve fluids of many different types. The heat transfer characteristics and temperature-pressure coupling vary with fluid type. Oil- and water-based liquids and polymers behave differently from compressible systems. Multiple fluids in the wellbore, including spacers and displacement fluids, are an important consideration. Temperature dependent properties must be updated as temperatures and rheological properties change with time and depth. Even with drilling muds, the viscosity changes with temperature during the mud’s circuit down the drillpipe and up the annulus, affecting the overall hydraulics of the system.

Flexibility in wellbore geometry is needed to accommodate different configurations such as deviated wells, liners, dual completions, and offshore risers. The geometry determines the cross-sectional flow area and the fluid velocity, which, in turn, governs the heat transfer. Temperatures during liner cementing are strongly influenced by the size of the liner and the annular clearance.

Flow types include production, injection, forward circulation, reverse circulation, drilling, and shut-in. Drilling is a special case of forward circulation, in which the depth of circulation and the wellbore thermal resistance change as the well is drilled and casing is set.

Static Wellbore Pressure Solutions

Static wellbore pressure solutions are the easiest to determine and are the most suitable for hand calculation. Because velocity is zero and no time dependent effects are present, we need only consider Eq. 3.2 with velocity terms deleted.

....................(3.9)

Temperatures are assumed to be static (often the undisturbed geothermal temperature) and known functions of measured depth.

Constant Density

The simplest version of Eq. 3.9 is the case of an incompressible fluid with constant density ρ.

....................(3.10)

where ΔZ is the change in true vertical depth (TVD) (i.e., hydrostatic head). For constant slope Φ, ΔZ equals cos Φ Δz. For a slightly compressible fluid, such as water, Eq. 3.9 could be used for small ΔZ increments where temperature and pressure values do not vary greatly.

Compressible Gas

To show a somewhat more complicated static pressure solution, consider the density equation for an ideal gas: , where T is absolute temperature, and R is a constant. For an ideal gas, density has an explicit dependence on pressure and temperature. The solution to Eq. 3.9 for a well with constant slope Φ is

....................(3.11)

where the initial condition for P is Po . T(z) is a given absolute temperature distribution, and z is the measured depth. For constant T, we see that the pressure of an ideal gas increases exponentially with depth, while an incompressible fluid pressure increases linearly with depth.

Flowing Wellbore Pressure Solutions

The next level of complexity in hydraulic calculations is the steady flow of the wellbore fluids. One part of this complexity is the calculation of the Fanning friction factor, f. This subject is postponed to the section on rheology, and f is assumed to be known in the following calculations.

Constant Density

The simplest version of Eq. 3.2 for flowing fluids is the case of an incompressible fluid with constant density ρ and rheological properties, and a constant-slope wellbore.

....................(3.12)

In this case, we have evaluated the Fanning friction factor from the appropriate equation in the rheological section. Note that Δv is zero through the mass conservation equation.

Linearly Varying Density

The next simplest solution has a linearly varying density along z. Conservation of mass requires that

....................(3.13)

where the o subscript indicates initial values, and no subscript indicates final values. With Eq. 3.13 , we can calculate Δv in terms of Δρ .

Eq. 3.15 may be used directly to calculate ΔP if the final density is insensitive to pressure. Otherwise, this equation must be solved numerically for the pressure. For instance, the density terms could be linearized with respect to P and Newton’s method used to converge to the final pressure.

Compressible Fluid

The flow of a compressible fluid can often produce results that seem counter to intuition. For example, consider the steady flow of air in a constant area duct. As with all fluids, there is a pressure loss because of friction, and the pressure decreases continuously from the entrance of the duct to the exit. Unlike the flow of incompressible fluids, the fluid velocity increases from the entrance of the duct to the exit. How could friction make the fluid speed up?

Two facts account for this acceleration. First, the gas pressure is proportional to the density (as in the ideal gas law P = ρRT). As the pressure of the gas decreases, the density must decrease also. Second, because the mass flow through the duct is constant, the product of density and velocity is constant. Thus, as the density decreases with the pressure, the velocity must increase to maintain the mass flow.

This example demonstrates a typical compressible flow characteristic—the interrelationship of pressure and mass flow. In air drilling, high velocities are needed at bottom of hole to remove the cuttings. High velocities result in friction pressure drops in the drillpipe and annulus, so higher standpipe pressures may be needed to keep the air flowing. Higher standpipe pressures result in higher gas densities and, hence, result in lower velocities. Fortunately, most air drilling operations do not result in the vicious circle situation previously described.

The pressure change in a flowing gas is properly a problem in gas dynamics. Gas dynamic analytic solutions are available for two cases of flow with friction: adiabatic and isothermal flow. Neither case is exactly what we need for these calculations, but the reader is directed to a gas dynamics reference for additional depth of understanding of this problem.[1] If we assume a linearly varying density and temperature, we can use the results of the previous section with the addition of a pressure/volume/temperature (PVT) relationship. For most applications, an ideal gas model is sufficiently accurate. With the relation P = ρRT, we can calculate ΔP in terms of density as

....................(3.16)

Substituting Eq. 3.16 into Eq. 3.15, we derive the quadratic equation for density.

....................(3.17)

If there are two positive roots for the density, the root that gives a subsonic velocity is the correct root. The speed of sound for an ideal gas is

....................(3.18)

where cp and cv are the heat capacities at constant pressure and volume, respectively.

General Steady Flow Wellbore Pressure Solutions

We make only one assumption in this general discussion of wellbore flow modeling, and that is that the temperature distribution is given. To make any other assumption requires a general solution of the energy equation for the wellbore, which is beyond the scope of this chapter. For review, we repeat the balance of mass and momentum for 1D flow along a constant area duct.

....................(3.19)

....................(3.20)

At changes of area, the following conditions hold.

....................(3.21)

and

....................(3.22)

Eq. 3.22 simplifies for incompressible flow.

....................(3.23)

where subscript o indicates inlet properties.

Given that we have a means of calculating the Fanning friction factor, we need a PVT relationship relating pressure, density, and temperature, and what is often available is a pressure function P(ρ,T) depending on density and temperature. When this is substituted into Eq. 3.20 , with Eq. 3.19 , we obtain the first-order differential equation in density.

....................(3.24)

Eq. 3.24 can be integrated numerically using any of several methods, such as adaptive Runge-Kutta or Bulirsh-Stoer. The reader is referred to numerical analysis books for details of these two methods.[2] Once the new density has been determined, the pressure can be calculated from the PVT relationship, and the velocity can be calculated from Eq. 3.19 . Alternately, we might have density as a function of pressure and temperature: ρ(P,T ). In this case, the differential equation is in terms of pressure.

....................(3.25)

Again, this is a first-order differential equation but now in terms of pressure. Once the pressure is determined, density is determined from the PVT relationship, and the density together with Eq. 3.19 gives the velocity.

Flow-area changes may act like nozzles (area reduction) or diffusers (area increases). The actual calculation of flow-property changes is beyond the capability of a 1D flow analysis. Often, we insert coefficients into the 1D equations to account for the complexity, and then we evaluate these coefficients from experiments. One such coefficient is the discharge coefficient shown in Eqs. 3.22 and 3.23 . A further comment is needed about the general Eq. 3.22 . Some assumption must be made about the variation of temperature with pressure within the nozzle before the integral can be evaluated. A typical assumption is that the flow is adiabatic (i.e., negligible heat transfer takes place in the nozzle). For our purposes, adiabatic is equivalent to isentropic, and entropy functions are available for many fluid models. For isentropic flow,

....................(3.26)

Or, we can derive the change of temperature with pressure.

....................(3.27)

For an ideal gas, we can solve Eq. 3.27 to get

....................(3.28)

where k = cp/cv . Using the ideal gas law, we can eliminate T.

....................(3.29)

We can now express pressure in terms of density in Eq. 3.22 and express v in terms of density with Eq. 3.21. The resulting equation can be solved numerically for density. Then, Eq. 3.29 can be used to determine the pressure.

Calculating Pressures in a Wellbore

Assuming we can calculate ΔP for each constant area section of drillpipe or annulus and can calculate ΔP for nozzles and area changes, we are now ready to evaluate the pressures in a wellbore. A typical wellbore fluid system is illustrated in Fig. 3.1.[3] Summing all pressure drops give the standpipe pressure.

In this calculation, we assume that the calculations are started from a known pressure value, most conveniently the atmospheric pressure at the exit of the annulus. This choice is particularly suitable if air or foam drilling is being considered because "choked" gas flow almost never occurs. For this choice of "boundary condition," flow calculations proceed backward from the annulus exit to the standpipe pressure. For flow in the annulus, both fluid density and fluid friction increase pressure going down the annulus. Where fluid type changes, the pressure and flow velocity are continuous.

and

....................(3.31)

Notice that mass flow rate may not be continuous at the interface between two fluids because the densities may be different: ρ1vA ≠ ρ2vA, where v and A are continuous at the interface. When calculating from the bit to the standpipe, inside the drillstring, fluid density decreases pressure and fluid friction increases pressure. Pressure changes because of internal upsets and tool joints consist of two area changes and a short flow section, as shown in Fig. 3.2.

Fig. 3.2—Pressure-drop calculation sections.

Pressure drop across the bit consists of two area changes: into the nozzles and exit from the nozzles into the openhole annular area. Miscellaneous pressure drops are drops through tools, mud motors, floats, or in-pipe chokes. Sometimes, the manufacturer will have this pressure-loss information tabulated; otherwise, you will have to estimate the pressure loss through use of the tool internal dimensions.

If the standpipe pressure is given, then the flow exiting the annulus must be choked back to atmospheric pressure.

....................(3.32)

Surge Pressure Prediction

Introduction

An exceptional flow case is the operation of running pipe or casing into the wellbore. Moving pipe into the wellbore displaces fluid, and the flow of this fluid generates pressures called surge pressures. When the pipe is pulled from the well, negative pressures are generated, and these pressures are called swab pressures. In most wells, the magnitude of the pressure surges is not critical because proper casing design and mud programs leave large enough margins between fracture pressures and formation-fluid pressures. Typically, dynamic fluid flow is not a consideration, so a steady-state calculation can be performed. A certain fraction of wells, however, cannot be designed with large surge-pressure margins. In these critical wells, pressure surges must be maintained within narrow limits. In other critical wells, pressure margins may be large, but pressure surges may still be a concern. Some operations are particularly prone to large pressure surges (e.g., running of low-clearance liners in deep wells). The reader is referred to papers on dynamic surge calculations,[4],[5] and a later section on dynamic pressure calculation gives a taste of this type of calculation.

The surge pressure analysis consists of two analytical regions: the pipe-annulus region and the pipe-to-bottomhole region (Fig. 3.3). The fluid flow in the pipe-annulus region should be solved using techniques already discussed, but with the following special considerations: frictional pressure drop must be solved for flow in an annulus with a moving pipe, and in deviated wells, the effect of annulus eccentricity should be considered. The analysis of the pipe-to-bottomhole region should consist of a static pressure analysis, with pressure boundary condition determined by the fluid flow at the bit, or pipe end if running casing. The pipe-annulus model and the pipe-to-bottomhole model then are connected through a comprehensive set of force and displacement compatibility relations.

Fig. 3.3—Surge-pressure calculation regions.

Boundary Conditions

The following conditions describe the flow for a surge or swab operation.

Surface Boundary Conditions.

There are six variables that can be specified at the surface:

P1 = pipe pressure.

v1 = pipe fluid velocity.

P2 = annulus pressure.

v2 = annulus fluid velocity.

v3 = pipe velocity.

A maximum of three boundary conditions can be specified at the surface. For surge without circulation, the following boundary conditions hold:

Surge Pressure Solution

Because of the complex boundary conditions, the solution of a steady-state surge pressure is most easily solved with a computer program. For closed-pipe and circulating cases, the flow is defined so that pressures can be calculated from the annulus exit to the standpipe, as discussed previously. For open-pipe surges, the problem is finding how the flow splits between the pipe and the annulus, so that the pressures for both the pipe and the annulus match at the bit. One strategy for solving this problem is given next.

1. Calculate all pressures with all flow in the annulus. Then, check pressures at the bit; annulus pressure will be lower because of fluid friction.

2. Calculate all pressures with all flow in the pipe. Then, check pressures at the bit; pipe pressure will be lower because of fluid friction.

3. Calculate a division of flow between the pipe and annulus that will equalize the pressures at the bit.

4. Repeat Step 3 until the two pressures match within an acceptable tolerance.

The efficiency of this calculation will depend on the method chosen for Step 3. With modern computers, this is not a particularly critical problem, so a simple interval halving technique would work. For the ith iteration of Step 3, fi is the fraction of flow in the pipe, and (1 – fi ) is the fraction in the annulus. Previous steps show that fp gives a higher annulus pressure and fm gives a lower annulus pressure. Our new choice for fi is ½(fp + fm). We perform the pressure calculation and find that the annulus pressure is higher, so we assign fp = fi . If the pressure difference is less than our tolerance, which we chose to be 1 psi, then the calculation is complete. Otherwise, we try another step. How do we establish fp and fm? The initial two steps in the solution step should give us fp = 0 and fm = 1, respectively. In some cases, such as small nozzles or restricted flow around the bit, fluid must flow into either the pipe or annulus, or the fluid level must fall. For these cases, f may be negative or greater than one. It may be necessary to repeat Steps 1 and 2 to establish the initial set fm and fp.

Fluid Friction

In the previous sections, we calculated the pressures in a flowing fluid, assuming we knew the value of the Fanning friction factor. Determination of the Fanning friction factor, in fact, may be the most difficult step in this calculation. Fluid friction is studied by the science of rheology.

Fluid Rheology

The science of rheology is concerned with the deformation of all forms of matter, but has had its greatest development in the study of the flow behavior of suspensions in pipes and other conduits. The rheologist is interested primarily in the relationship between flow pressure and flow rate, and in the influence thereon of the flow characteristics of the fluid. There are two fundamentally different relationships:

The laminar flow regime prevails at low flow velocities. Flow is orderly, and the pressure-velocity relationship is a function of the viscous properties of the fluid.

The turbulent flow regime prevails at high velocities. Flow is disorderly and is governed primarily by the inertial properties of the fluid in motion. Flow equations are empirical.

The laminar flow equations relating flow behavior to the flow characteristics of the fluid are based on certain flow models, namely the Newtonian, the Bingham plastic, the pseudoplastic, the yield power-law, and the dilatant. Only the first four are of interest in drilling-fluid technology. Most drilling fluids do not conform exactly to any of these models, but drilling-fluid behavior can be predicted with sufficient accuracy by one or more of them. Flow models are usually visualized by means of consistency curves, which are plots either of flow pressure vs. flow rate or of shear stress vs. shear rate.

Shear stress is force per unit area and is expressed as a function of the velocity gradient of the fluid as

....................(3.33)

where μ is the fluid viscosity and dv/dr is the velocity gradient. The negative sign is used in Eq. 3.33 because momentum flux flows in the direction of negative velocity gradient. That is, the momentum tends to go in the direction of decreasing velocity The absolute value of velocity gradient is called the shear rate and is defined as

....................(3.34)

Then, Eq. 3.33 can be written as

....................(3.35)

Viscosity is the resistance offered by a fluid to deformation when it is subjected to a shear stress. If the viscosity is independent of the shear rate, the fluid is called a Newtonian fluid. Water, brines, and gases are examples of Newtonian fluid. The shear stress is linear with the shear rate for a Newtonian fluid and is illustrated by Curve A in Fig. 3.6. The symbol μ without any subscript is used to refer to the viscosity of Newtonian fluid. Most of the fluids used in drilling and cementing operations are not Newtonian, and their behavior is discussed next.

Fig. 3.6—Rheology of fluids.

If the viscosity of a fluid is a function of shear stress (or, equivalently, of shear rate), such a fluid is called non-Newtonian fluid. Non-Newtonian fluids can be classified into three general categories:

Fluid properties are independent of duration of shear.

Fluid properties are dependent on duration of shear.

Fluid exhibits many properties that are characteristics of solids.

Time Independent.

The following three types of materials are in this class.

Bingham Plastic. These fluids require a finite shear stress, τy; below that, they will not flow. Above this finite shear stress, referred to as yield point, the shear rate is linear with shear stress, just like a Newtonian fluid. Bingham fluids behave like a solid until the applied pressure is high enough to break the sheer stress, like getting catsup out of a bottle. The fluid is illustrated by Curve B in Fig. 3.6. The shear stress can be written as

....................(3.36)

where τy is called the yield point (YP), and μp is referred to as the plastic viscosity (PV) of the fluid. Some water-based slurries and sewage sludge are examples of Bingham plastic fluid. Most of the water-based cement slurries and water-based drilling fluids exhibit Bingham plastic behavior. Drilling muds are often characterized with YP and PV values, but this is for historical reasons and does not necessarily imply that the Bingham fluid model is the best model for all muds.

Pseudoplastic. These fluids exhibit a linear relationship between shear stress and shear rate when plotted on a log-log paper. This is illustrated by Curve C in Fig. 3.6. This fluid is also commonly referred to as a power-law non-Newtonian fluid. The shear stress can be written as

....................(3.37)

where K is the consistency index, and n is the exponent, referred to as power-law index. A term μa is defined that is called the apparent viscosity and is

....................(3.38)

Note that apparent viscosity and effective viscosity as defined by different authors are not always defined in the sense used here, so read with caution. The apparent viscosity decreases as the shear rate increases for power-law fluids. For this reason, another term commonly used for pseudoplastic fluids is "shear thinning." Polymeric solutions and melts are examples of power-law fluid. Some drilling fluids and cement slurries, depending on their formulation, may exhibit power-law behavior.

Yield Power Law. Also known as Herschel-Bulkley fluids, these fluids require a finite shear stress, τy, below which they will not flow. Above this finite shear stress, referred to as yield point, the shear rate is related to the shear stress through a power-law type relationship. The shear stress can be written as

....................(3.39)

where τy is called the yield point, K is consistency index, and m is the exponent, referred to as power-law index.

Dilatant.

These fluids also exhibit a linear relationship between shear stress and shear rate when plotted on a log-log paper and are illustrated as Curve D in Fig. 3.6. The shear stress expression for dilatant fluid is similar to power-law fluid, but the exponent n is greater than 1. The apparent viscosity for these fluids increases as shear rate increases. For this reason, dilatant fluids are often called "shear-thickening."

Quicksand is an example of dilatant fluid. In cementing operations, it would be disadvantageous if fluids increased in viscosity as shear stress increased.

Time Dependent. These fluids exhibit a change in shear stress with the duration of shear. This does not include changes because of reaction, mechanical effects, etc. Cement slurries and drilling fluids usually do not exhibit time-dependent behavior. However, with the introduction of new chemicals on a regular basis, one should test and verify the behavior.

Solids Characteristic. These fluids exhibit elastic recovery from deformation that occurs during flow and are called viscoelastic. Most of the cement slurries and drilling fluids do not exhibit this behavior. However, as mentioned earlier, new polymers are being introduced on a regular basis, and tests should be conducted to verify the behavior.

The unit of viscosity is Pascal-second (Pa-s) in the SI system and lbf/(ft-s) in oilfield units. One Pa-s equals 10 poise (P), 1,000 centipoise (cp), or 0.672 lbf/(ft-s). The exponent n is dimensionless, and consistency index, K, has the units of Pa-sn in the SI system and lbf/(secn-ft–2) in oilfield units. One Pa-sn equals 208.86 lbf/(secn.ft–2). The yield point for Bingham fluids is often characterized in units of lbf/(1,00ft2), and plastic viscosity is usually given in centipoise.

Viscometry.

The rheology parameters of the fluids, μ and μp , and τo , K, and n, are determined by conducting tests in a concentric viscometer. This consists of concentric cylinders with one of them rotating, usually the outer one. A sample of fluid is placed between the cylinders, and the torque on the inner cylinder is measured. Assuming an incompressible fluid, with flow in the laminar flow regime, the equations of motion can be solved for τ to give

....................(3.40)

where

τ = shear stress, Pa;

MT = torque, N-m;

L = length, m;

and

r = radius, m.

In a concentric viscometer, torque, MT, is measured at a different rotational speed of the outer cylinder. Shear stress is then calculated from Eq. 3.40 , and shear rate is given by

and

....................(3.41)

where

Rb = radius of inner cylinder (bob), m;

Rc = radius of outer cylinder (cup), m;

κ = ratio of radius of inner cylinder to outer cylinder;

and

Ω0 = angular velocity of outer cylinder.

Shear stress and shear rate are then analyzed to determine the rheology model.

A number of commercially available concentric cylinder rotary viscometers are suitable for use with drilling muds. They are similar in principle to the viscometer already discussed. All are based on a design by Savins and Roper, which enables the plastic viscosity and yield point to be calculated very simply from two dial readings, at 600 and 300 rpm, respectively.[6] They are referred to in the industry as the direct-indicating viscometer and typically are called Fann viscometers.

The underlying theory is as follows: Eqs. 3.40 and 3.41 are combined to give

....................(3.42)

where avs and bvs are constants that include the instrument dimensions, the spring constant, and all conversion factors; ω is the rotor speed in revolutions per minute (rpm).

Then,

....................(3.43)

where θ1 and θ2 are dial readings taken at ω1 and ω2 rpm, respectively. PV is the conventional oilfield term for plastic viscosity, thus measured. Then, the yield point is determined.

....................(3.44)

YP is the conventional oilfield term for yield point, thus measured. The numerical values of avs , bvs , ω1 , and ω2 were chosen so that

....................(3.45)

and

....................(3.46)

Apparent viscosity μa may be calculated from the Savins-Roper viscometer reading as

and

....................(3.47)

where θ is the dial reading at ω rpm. Typical viscometer results are shown in Fig. 3.7.[7] Notice that real fluids are not ideally any of the models shown, but generally are pretty close to one model or another. The selection of the model may be motivated by a particular fluid velocity of interest. For instance, fluid 6 in Fig. 3.7 would be modeled well by a yield-power law for rpm below about 100.

Fanning Friction Factor Correlations.
Flow in pipes and annuli are typically characterized as laminar or turbulent flow. Laminar flow often can be solved analytically. Correlation for turbulent flow is usually developed empirically by conducting experiments in a flow loop. Typical data will look like those that are shown in Fig. 3.8. Experimental data are usually analyzed and correlated through the use of two dimensionless numbers: f, the Fanning friction factor, and Re, the Reynolds number. The relationship between the friction factor, f, and Reynolds number for Newtonian fluids is given in Fig. 3.9,[8] with the pipe roughness given in Fig. 3.10. This figure is based on the experimental results of Colebrook.[9] The relationship between friction factor f vs. Re for pseudoplastic fluids is shown in Fig. 3.11. This figure is based on the experimental results of Dodge and Metzner.[10] Here non-Newtonian fluids usually assume this pseudoplastic friction factor for turbulent flow.

where f is Fanning friction factor, Δz is the length, v is the velocity, ρ is the density, Dhyd is a characteristic "diameter," and ΔP is the pressure drop. The friction factor depends on Reynolds number, Re, and the roughness of the pipe. The Reynolds number, Re, is defined as

....................(3.49)

where ρ is the density of the fluid, v is the average velocity, D is a characteristic length (e.g., pipe diameter), and μ is a characteristic viscosity. Correlations for friction factor, f, in both laminar and turbulent flow regime and for critical Reynolds number are available for a number of fluids and geometries. However, in critical situations, it is recommended that flow-loop tests be conducted and data compared with calculations that are based on fundamental equations for flow. For example, experimental data in laminar flow should be compared with estimated values from correlation such as Eq. 3.52. However, some solid-laden polymers are known to exhibit what is known as shear-induced diffusion, in which solids migrate away from the walls to the center of the pipe. These fluids show deviation in calculated and experimental values in laminar flow. Correlations should be modified as needed to reflect this behavior. Several polymers are known to exhibit drag reduction in turbulent flow. Theoretical prediction of polymer-flow behavior is not yet good enough, so flow-loop data are almost always needed.

Commonly used Fanning friction correlations are summarized in the next section. Correlations are provided for three geometric configurations: pipe flow, concentric annular flow, and slit flow. For each case, the ΔP and Re are defined for the specific geometry and flow model. The laminar flow equations for annular flow are approximate for Newtonian and power-law flow in annuli with low-clearance but reasonably accurate and much simpler than the exact solutions. Note that for low clearance annuli, the slit flow model provides almost as accurate a result as the concentric annular model but can also be modified to account for eccentric annuli.

Rheological Model 1: Newtonian Fluids

Pipe Flow.

Frictional pressure drop:

....................(3.50)

Reynolds number:

....................(3.51)

where Di is the pipe inside diameter (ID).

Laminar flow:

....................(3.52)

for Re < 2,100.

Turbulent flow:

....................(3.53)

for Re > 3,000, and k is the absolute pipe roughness in the same units as D.

Annular Flow.

Frictional pressure drop:

....................(3.54)

Reynolds number:

....................(3.55)

where Do is the annulus outside diameter (OD), and Di is the ID.

Laminar flow:

....................(3.56)

for Re < 2,100.

Turbulent flow:

....................(3.57)

for Re > 3,000, and k is the absolute pipe roughness in the same units as D.

Rheological Model 2: Bingham Plastic Fluids

Pipe Flow.

Frictional pressure drop:

....................(3.58)

Reynolds number:

....................(3.59)

where Di is the pipe ID, and μp is the plastic viscosity.

Laminar flow:

....................(3.60)

for Re < ReBP1, where

Turbulent flow:

....................(3.61)

for Re > ReBP2 , where:

For He < = 0.75 × 105 , A = 0.20656, and B = 0.3780.

For 0.75 × 10 5 < He < = 1.575 × 10 5 , A = 0.26365, and B = 0.38931.

For He > 0.75 × 10 5 , A = 0.20521, B = 0.35579, and He = τoρD2/μp2 .

Annular Flow.

Frictional pressure drop:

....................(3.62)

Reynolds number:

....................(3.63)

where Do is the annulus OD; Di is the ID; and μp is the plastic viscosity.

The frictional pressure drop in an eccentric annulus is known to be less than the frictional pressure drop in a concentric annulus. For laminar flow of Newtonian fluids, the pressure drop in a fully eccentric annulus is half the pressure drop in a concentric annulus. For turbulent flow, the difference is about 10%. For non-Newtonian fluids, the effect is less but still significant. In deviated wells, the drillpipe should be fully eccentric over much of the deviated wellbore, resulting in reduced fluid friction.

Ce for laminar flow is determined based on the methods used by Uner et al.[11] The flow rate through a concentric annulus is given by

....................(3.92)

where Rr = ri/ro . The flow rate through an eccentric annulus was determined to be

....................(3.93)

where

....................(3.94)....................(3.95)

and

....................(3.96)

where δr is the distance between centers of the inside and outside pipes (e.g., δr = 0 for concentric pipes). The geometry of the eccentric annulus is illustrated in Fig. 3.12.

Fig. 3.12—Annulus eccentricity.

The function E may be evaluated using a six-coefficient approximation. The function F must be evaluated using numerical methods (e.g., a seven-point Newton-Cotes numerical integration formula). Setting qa and qe equal, then

....................(3.97)

Because Ce depends only on f, n, and Rr, Ce need be calculated only once, then used for all future frictional pressure drop calculations, as long as the property n does not vary.

Ce for turbulent flow is determined by applying the same techniques to the turbulent velocity profile determined by Dodge and Metzner.[10]

The integral in Eq. 3.105 must be evaluated numerically (e.g., by a seven-point Newton-Cotes numerical integration. Ce can be determined by setting Eq. 3.104 equal to Eq. 3.105 and noting that

....................(3.108)

where vc* is determined from the concentric solution given by Dodge and Metzner.[10] The resulting nonlinear equation must be solved for Ce numerically (e.g., by using Newton’s method). Because Ce depends only on f1, f2 , f3 , n, and Rr , Ce need be calculated only once, then used for all future frictional pressure-drop calculations, as long as the properties ρ,K, and n do not vary.

Dynamic Pressure Prediction

Introduction

Calculating dynamic pressures in a wellbore are significantly more difficult than calculating steady-state flowing conditions. In a dynamic calculation, there are two effects not considered in steady flow: fluid inertia and fluid accumulation. In steady-state mass conservation, flow of fluid into a volume was matched by an equivalent flow out of the volume. In the dynamic calculation, there may not be equal inflow and outflow, but instead, fluid may accumulate within the volume. For fluid accumulation to occur, either the fluid must compress or the wellbore must expand. When considering the momentum equation, the fluid at rest must be accelerated to its final flow rate. The fluid inertia resists the change in velocity.

Typically, dynamic fluid flow is not a consideration. One exception is the operation of running pipe or casing into the wellbore, where dynamic pressure variation may be as important as pressures because of fluid friction. A second area of interest might be water-hammer effects during production startup.

Governing Equations—Dynamic Pressure Prediction

The fluid pressures and velocities in open hole are determined by solving two coupled partial differential equations: the balance of mass and the balance of momentum.

Balance of Mass

....................(3.109)

where

A = cross-sectional area, m2;

P = pressure, Pa;

Kb = fluid bulk modulus, Pa;

and

v = fluid velocity, m/s.

The term

....................(3.110)

is the compressibility, C, of the wellbore/fluid system (i.e., the change in wellbore volume per unit change in pressure). The balance of mass consists of three effects: the expansion of the hole because of internal fluid pressure, the compression of the fluid because of changes in fluid pressure, and the influx or outflux of the fluid. The expansion of the hole is governed by the elastic response of the formation and any casing cemented between the fluid and the formation. The fluid volume change is given by the bulk modulus K. For drilling muds, K varies as a function of composition, pressure, and temperature. The reciprocal of the bulk modulus is called the compressibility.

Balance of Momentum

....................(3.111)

where

ρ = fluid density, kg/m3;

f = Fanning friction factor;

Dh = wellbore diameter, m;

g = gravitational constant, m/s2;

Φ = angle of inclination from the vertical;

and

The balance of momentum equation consists of four terms. The first term in Eq. 3.111 represents the inertia of the fluid [i.e., the acceleration of the fluid (left side of Eq. 3.111 ) equals the sum of the forces on the fluid (right side of Eq. 3.111 )]. The last three terms are the forces on the fluid. The first of these terms is the pressure gradient. The second is the drag on the fluid because of frictional or viscous forces. The friction factor f is a function of the type of fluid and the velocity of the fluid. Frictional drag is discussed in the section on rheology. The last force is the gravitational force.

The balance equations for flow with a pipe in the wellbore are similar to the equations for the openhole model with two important differences. First, the expansivity terms in the balance of mass equations depend on the pressures both inside and outside the pipe. For instance, increased annulus pressure can decrease the cross-sectional area inside the pipe, and increased pipe pressure can increase the cross-sectional area because of pipe elastic deformation. The second major difference is the effect of pipe speed on the frictional pressure drop in the annulus, as discussed in the steady-state surge article. Consult papers on dynamic surge pressures for more detail concerning the wellbore/pipe problem.[4],[5]

Borehole Expansion

The balance of mass equation contains a term that relates the flow cross-sectional area to the fluid pressures. This section discusses the application of elasticity theory to the determination of the coefficients in the balance of mass equation. If we assume that the formation outside the wellbore is elastic, then the displacement of the borehole wall because of change in internal pressure is given by the elastic formula.

....................(3.112)

where

u = radial displacement, m;

υf = Poisson’s ratio for the formation;

and

Ef = Young’s modulus for the formation, Pa.

The cross-sectional area of the annulus is given by

....................(3.113)

If we assume u is small compared to D, we can calculate the following formula from Eqs. 3.112 and 3.113.

....................(3.114)

Using typical values of formation elastic modulus, the borehole expansion term is the same order of magnitude as the fluid compressibility and cannot be neglected.

Solution Method—Fluid Dynamics

The method of characteristics is the method most commonly used to solve the dynamic pressure-flow equations. This method has been extensively used in the analysis of dynamic fluid flow. However, applying the method of characteristics to realistic wellbore flow problems has the following difficulties:

Iteration may be necessary to solve for characteristics and flow variables when properties and geometry vary in space.

Multiple coordinate systems must be computed and related to a fixed coordinate system.

Interpolation is necessary when characteristic curves do not intersect the spatial point of interest.

Moving coordinate systems must be continually updated so that only points within the fixed-coordinate system are computed.

These difficulties can be reduced or eliminated by using the following approach:

Adopt a fixed spatial grid.

For a given time step, integrate the characteristic curves and flow equations from each gridpoint. Note that the flow equations are now evaluated at the new spatial point obtained from the characteristic curves.

Interpolate the flow equations back to the fixed grid and solve for the flow variables.

This method eliminates the moving coordinate systems and replaces them with a set of interpolation factors. Because the grid is fixed, fluid properties and well geometry are known at each gridpoint, and no iteration is necessary. Most of the equations can be "presolved" so that they only need to be evaluated at each timestep. The disadvantages of this method are that the fluid variables must be evaluated at each gridpoint rather than only at points of interest, and that a maximum timestep size is required for stability.

The characteristic equations are developed using the methods given in Chap. 1 of Lapidus and Pindar.[12] For the open hole below the moving pipe, the fluid motion is governed by the system of equations shown in Eq. 3.115.

....................(3.115)

where the first two equations are the balance of mass, with C equal to the wellbore-fluid compressibility, and the balance of momentum, with friction and gravitation terms lumped together as h.

....................(3.116)

The last two equations describe the variation of p and v along the characteristic curve ξ = z ± at, where a is the acoustic velocity. Subscripts here denote partial derivatives (e.g., vz∂v/∂z). This system of equations is overdetermined; that is, there are more equations than unknowns. For this system to have a solution, the following condition must hold.

....................(3.117)

Evaluating the determinant (Eq. 3.117) defines the acoustic velocity.

....................(3.118)

The second condition that the equations have a solution requires

....................(3.119)

This determinant produces the following differential equations along the characteristic curve.

....................(3.120)

The characteristic equations are solved to give p(x,t) and v(x,t) in the following way. Eq. 3.120 is integrated along the characteristics for time step Δt.

....................(3.121)

and

....................(3.122)

Eqs. 3.121 and 3.122 can be solved simultaneously to give

....................(3.123)

and

....................(3.124)

Generally, c+ and c– must be interpolated to give values at the points of interest.[13]

Cuttings Transport

Introduction

Of the many functions that are performed by the drilling fluid, the most important is to transport cuttings from the bit up the annulus to the surface. If the cuttings cannot be removed from the wellbore, drilling cannot proceed for long. In rotary drilling operations, both the fluid and the rock fragments are moving. The situation is complicated further by the fact that the fluid velocity varies from zero at the wall to a maximum at the center of pipe. In addition, the rotation of the drillpipe imparts centrifugal force on the rock fragments, which affects their relative location in the annulus. Because of the extreme complexity of this flow behavior, drilling personnel have relied primarily on observation and experience for determining the lifting ability of the drilling fluid. In practice, either the flow rate or effective viscosity of the fluid is increased if problems related to inefficient cuttings removal are encountered. This has resulted in a natural tendency toward thick muds and high annular velocities. However, increasing the mud viscosity or flow rate can be detrimental to the cleaning action beneath the bit and cause a reduction in the penetration rate. Thus, there may be a considerable economic penalty associated with the use of a higher flow rate or mud viscosity than necessary. Transport is usually not a problem if the well is near vertical. However, considerable difficulties can occur when the well is being drilled directionally, because cuttings may accumulate either in a stationary bed at hole angles above about 50° or in a moving, churning bed at lower hole angles. Drilling problems that may result include stuck pipe, lost circulation, high torque and drag, and poor cement jobs. The severity of such problems depends on the amount and location of cuttings distributed along the wellbore.

Vertical Wells

The problem of cuttings transport in vertical wells has been studied for many years, with the earliest analysis of the problem being that of Pigott.[14] Several authors have conducted experimental studies of drilling-fluid carrying capacity. Williams and Bruce were among the first to recognize the need for establishing the minimum annular velocity required to lift the cuttings.[15] In 1951, they reported the results of extensive laboratory and field measurements on mud carrying capacity. Before their work, the minimum annular velocity generally used in practice was about 200 ft/min. As a result of their work, a value of about 100 ft/min gradually was accepted. More recent experimental work by Sifferman and Becker indicates that while 100 ft/min may be required when the drilling fluid is water, a minimum annular velocity of 50 ft/min should provide satisfactory cutting transport for a typical drilling mud. [16],[17]

The transport efficiency in vertical wells is usually assessed by determining the settling velocity, which is dependent on particle size, density and shape; the drilling fluid rheology and velocity; and the hole/pipe configuration. Several investigators have proposed empirical correlations for estimating the cutting slip velocity experienced during rotary-drilling operations. While these correlations should not be expected to give extremely accurate results for such a complex flow behavior, they do provide valuable insight in the selection of drilling-fluid properties and pump-operating conditions. The correlations of Moore, Chien, and Walker and Mayes have achieved the most widespread acceptance.[18]

Deviated Wells

Since the early 1980s, cuttings transport studies have focused on inclined wellbores. And an extensive body of literature on both experimental and modeling work has developed. Experimental work on cuttings transport in inclined wellbores has been conducted using flow loops at the U. of Tulsa and elsewhere. Different mechanisms, which dominate within different ranges of wellbore angle, determine cuttings bed heights and annular cuttings concentrations as functions of operating parameters (flow rate and penetration rate), wellbore configuration (depth, hole angle, hole size or casing ID, and pipe size), fluid properties (density and rheology), cuttings characteristics (density, size, bed porosity, and angle of repose), and pipe eccentricity and rotary speed.

Laboratory experience indicates that the flow rate, if high enough, will always remove the cuttings for any fluid, hole size, and hole angle. Unfortunately, flow rates high enough to transport cuttings up and out of the annulus effectively cannot be used in many wells because of limited pump capacity and/or high surface or downhole dynamic pressures. This is particularly true for high angles with hole sizes larger than 12¼ in. High rotary speeds and backreaming are often used when flow rate does not suffice.

Particle Slip Velocity

The earliest analytical studies of cuttings transport considered the fall of particles in a stagnant fluid, with the hope that these results could be applied to a moving fluid with some degree of accuracy. Most start with the relation developed by Stokes for creeping flow around a spherical particle.[19]

....................(3.125)

where

μ = Newtonian viscosity of the fluid, Pa-s;

ds = particle diameter, m;

vsl = particle slip velocity, m/s;

and

Fd = total viscous drag force on the particle, N.

When the Stokes drag is equated to the buoyant weight of the particle W,

....................(3.126)

Then, the slip velocity is given by

....................(3.127)

where

ρs = solid density, kg/m3 ;

ρf = fluid density, kg/m3 ;

and

g = acceleration of gravity, m/s2 .

Stokes’ law is accurate as long as turbulent eddies are not present in the particle’s wake. The onset of turbulence occurs for

....................(3.128)

where the particle Reynolds number is given by

....................(3.129)

For turbulent slip velocities, the drag force is given by

....................(3.130)

where f is an empirically determined friction factor. The friction factor is a function of the particle Reynolds number and the shape of the particle given by Ψ, the sphericity. Table 3.1 gives the sphericity of various particle shapes.

Table 3.1

The friction factor/Reynolds number relationship is shown in Fig. 3.13 for a range of sphericity. The particle slip velocity for turbulent flow is given by

If we define a laminar friction factor, f = 24/Rep, then Eq. 3.131 is valid for all Reynolds numbers.

Non-Newtonian fluids introduce new factors into particle-settling calculations. For a Bingham fluid, the particle will remain suspended with no settling if

....................(3.132)

where

τy is the fluid YP. Otherwise, because no other analytic solutions exist, an "apparent" or "equivalent" viscosity is determined from the non-Newtonian fluid parameters. For example, Moore used the apparent viscosity proposed by Dodge and Metzner for a pseudoplastic fluid.[17]

where μp is the plastic viscosity. The apparent viscosity models with most widespread acceptance are those of Moore.[21]

Carrying Capacity of a Drilling Fluid for Vertical Wells

The cuttings slip velocity is used to specify the minimum flow rate needed to clean the wellbore. This determination is not as straightforward as one might expect. In rotary-drilling operations, both the fluid and the rock fragments are moving. The situation is complicated further by the fact that the fluid velocity varies from zero at the wall to a maximum at the center of annulus. In addition, the rotation of the drillpipe imparts centrifugal force to the rock fragments, which affects their relative location in the annulus. Because of the extreme complexity of this flow behavior, drilling personnel have relied primarily on observation and experience for determining the lifting ability of the drilling fluid. In practice, either the flow rate or effective viscosity of the fluid is increased if problems related to inefficient cuttings removal are encountered. This has resulted in a natural tendency toward thick muds and high annular velocities. However, increasing the mud viscosity or flow rate can be detrimental to the cleaning action beneath the bit and cause a reduction in the penetration rate. Thus, there may be a considerable economic penalty associated with the use of a higher flow rate or mud viscosity than necessary.

As stated in the previous section, several investigators have proposed empirical correlations for estimating the cutting slip velocity experienced during the drilling process. While these correlations are not extremely accurate, they do give useful qualitative information about the cuttings transport process in vertical wells.

Five Percent Maximum Concentration Model for Vertical Wells

The following model was taken from Clark and Bickham.[22] For vertical well conditions, Fig. 3.14 shows a schematic of the cuttings transport process in a YPL fluid under laminar flow conditions. The area open to flow is characterized as a tube instead of an annulus. This simplifies the wellbore geometry. The tube diameter is based on the hydraulic diameter for pressure-drop calculations.

Fig. 3.14—Cuttings velocity profiles: YPL fluid.

Because drilling mud often exhibits a yield stress, there may be a region, near the center of the cross section, where the shear stress is less than the yield stress. There, the mud will move as a plug (i.e., rigid body motion). The plug velocity is vp. The average cuttings concentration and velocity in the plug are cp and vcp, respectively. In the annular region around the plug, the mud flows with a velocity gradient and behaves as a viscous fluid. The average annular velocity of the mud in this region is va. In addition, for the cuttings in this region, the average concentration and velocity are Ca and vca, respectively.

Cross-Sectional Geometry

First, let us define the basic wellbore geometry. The hydraulic diameter is defined as four times the flow area divided by the length of the wetted perimeter; namely,

....................(3.135)

For the wellbore annulus, the hydraulic diameter of the wellbore cross section is

....................(3.136)

where Dh is the wellbore diameter, and Dp is the drillpipe OD. The equivalent diameter is defined as

....................(3.137)

where A is the area open to flow. For the wellbore annulus, the equivalent diameter is

....................(3.138)

The plug diameter ratio is

....................(3.139)

Flow Conditions

The mixture velocity is

....................(3.140)

where Qm is the volumetric flow rate of the mud and Qc is the volumetric flow rate of the cuttings, which depends on the bit size and the penetration rate. In addition, the mixture velocity can be calculated from the average plug and annulus velocities in the equivalent pipe; namely,

....................(3.141)

Cuttings Concentration

The feed concentration is defined as

....................(3.142)

The average concentration, c, of cuttings in a short segment with length, Δz, and cross-sectional area, A, can be calculated as

....................(3.143)

The cuttings concentrations in the plug and annular regions are assumed equal. This means that the suspended cuttings are uniformly distributed across the area open to flow. Obviously, this assumption has a major impact, and the actual distribution is probably a function of wellbore geometry, mud properties, cuttings properties, and operating conditions. Thus, we obtain

....................(3.144)

where

....................(3.145)

is the average settling velocity in the axial direction. The components of the settling velocities in the axial direction are

....................(3.146)

and

....................(3.147)

where

and

....................(3.148)

CD is the drag coefficient of a sphere, τy is the yield stress of the mud, and μa is the apparent viscosity of the mud at a shear rate resulting from the settling cutting.

The value calculated using Eq. 3.144 is the minimum acceptable mixture velocity required for a cuttings concentration, c. Pigott[14] recommended that the concentration of suspended cuttings be a value less than 5%. With this limit (c = 0.05), Eq. 3.144 becomes

....................(3.149)

where co < 0.05. This implies that the penetration rate must be limited to a rate that satisfies this equality.

For near-vertical cases, the critical mud-cuttings mixture velocity equals the value of Eq. 3.149. If the circulation rate exceeds this value, the suspended cuttings concentration will remain less than 5%. However, if the mud circulation velocity is less than the cuttings’ settling velocity, the cuttings will eventually build up in the wellbore and plug it.

Cuttings Transport in Deviated Wells

A comprehensive cuttings transport model should allow a complete analysis for the entire well, from surface to the bit. The different mechanisms which dominate within different ranges of wellbore angle should be used to predict cuttings bed heights and annular cuttings concentrations as functions of operating parameters (flow rate and penetration rate), wellbore configuration (depth, hole angle, hole size or casing ID, and pipe size), fluid properties (density and rheology), cuttings characteristics (density, size, bed porosity, and angle of repose), pipe eccentricity, and rotary speed. Because of the complexity, extensive experimental data were necessary to help formulate and validate the new cuttings transport models.

New Experimental Data

Large-scale cuttings transport studies in inclined wellbores were initiated at the Tulsa U. Drilling Research Projects (TUDRP) in the 1980s with the support of major oil and service companies. A flow loop was built that consisted of a 40-ft length of 5-in. transparent annular test section and the means to vary and control

The angles of inclination between vertical and horizontal.

Mud pumping flow rate.

Drilling rate.

Drillpipe rotation and eccentricity.

Tomren et al.[23] found marked difference between the cuttings transport in inclined wellbores and that of vertical wellbores. A cuttings bed was observed to form at inclination angles of more than 35° from vertical, and this bed could slide back down for angles up to 50°. Eccentricity, created by the drillpipe lying on the low side of the annulus, was found to worsen the situation. Analysis of annular fluid flow showed that eccentricity diverts most of the mud flow away from the low side of the annulus, where the cuttings tend to settle, to the more open area above the drillpipe. Okrajni and Azar[24] investigated the effect of mud rheology on hole cleaning. They observed that removing a cuttings bed with a high-viscosity mud, a remedy for the hole-cleaning problem in vertical wells, may in fact be detrimental in high-angle wellbores (assuming a zero to low drillpipe rotation) and that a low-viscosity mud that can promote turbulence is more helpful. On the basis of this finding and on the previous study, hole cleaning was found to depend on the angle of inclination, hydraulics, mud rheological properties, drillpipe eccentricity, and rate of penetration. Becker et al.[25] then showed that the cuttings transport performance of the muds tested correlated best with the low-end-shear-rate viscosity, particularly the 6-rpm Fann V-G viscometer dial readings.

By the mid-1980s, a general qualitative understanding of the hole-cleaning problem in highly inclined wellbores had been gained. Because more directional and horizontal wells with longer lateral reaches were being drilled, the need for more and new experimental data created a demand for additional flow loops. In partnership with Chevron, Conoco, Elf Aquitaine, and Philips, TUDRP built a new and larger flow loop, with a 100-ft-long test section of 8-in. annulus, while construction of new flow loops was also done at Heriot-Watt U., BP, Southwest Research, M.I. Drilling Fluids, and the Inst. Français du Pétrole. All the flow loops had a transparent part of the annular test section that allowed observation of the cuttings transport mechanism. These flow loops provided the necessary tools for collecting the badly needed experimental data.

Because of the new flow loops, a significant amount of experimental data was collected on the effect of different parameters on cuttings transport under various conditions. The observations made and subsequent analysis of the data collected provided the basis for work toward formulating correlations/models.

Larsen conducted extensive studies on cuttings transport, totaling more than 700 tests with the TUDRP’s 5-in. flow loop. Tests were performed for angles from vertical to horizontal under critical as well as subcritical flow conditions. Critical flow corresponds to the minimum annular average fluid velocity that would prevent stationary accumulation of cuttings bed. Subcritical flow refers to the condition where a stationary cuttings bed forms. Analysis of the experimental data shows that when the fluid velocity is below the critical value, a cuttings bed starts to form and grows in thickness until the fluid velocity above the bed reaches the critical value. The critical velocity was reported in the range of 3 to 4 ft/sec, depending on the value of various parameters, such as the mud rheology, drilling rate, pipe eccentricity, and rotational speed. There were several new findings:

Under subcritical flow conditions, a medium-rheology mud (PV = 14 and YP = 14) consistently resulted in slightly smaller cuttings beds than those obtained with the low-rheology (PV = 7 and YP = 7) or the high-rheology (PV = 21 and YP = 21) muds. Calculation of the Reynolds number for the tests suggests that the flow regime for this mud is neither turbulent nor laminar but in the transition range.

The small cuttings size used (0.1 in.) in the study was more difficult to clean than the medium (0.175 in.) and the large (0.275 in.) sizes (drillpipe rpm 0 to 50). The small cuttings formed a more packed and smooth bed.

The height of the cuttings bed between 55 and 90° remained about the same, but there was a slight increase at about 65 to 70°.

Significant backsliding of the cuttings bed was observed for angles from 35 to 55°.

Seeberger et al.[26] reported that elevating the low shear rate viscosities enhances the cuttings-transport performance of oil muds. Sifferman and Becker[17] conducted a series of hole-cleaning experiments in an 8-in. flow loop. Statistical analysis of the data showed interaction among various parameters; thus, simple relationships could not be derived. For example, the effect of drillpipe rotation on cuttings transport depended also on the size of the cuttings and the mud rheology. The effect of rotation was more pronounced for smaller particles and for more viscous muds. Bassal[27] completed a study of the effect of drillpipe rotation on cuttings transport in inclined wellbores. The variables considered in this work were drillpipe rotary speed, hole inclination, mud rheology, cuttings size, and mud flow rate. Results have shown that drillpipe rotation has a significant effect on hole cleaning in directional well drilling. The level of enhancement in cuttings removal as a result of rotary speed is a function of a combination of mud rheology, cuttings size, mud flow rate, and the manner in which the drillstring behaves dynamically.

New Cuttings Transport Models

Larsen et al.[28] developed a model for highly inclined (50 to 90° angle) wellbores. The model predicts the critical velocity as well as the cuttings-bed thickness when the flow rate is below that of the critical flow. Hemphill and Larsen[29] showed that oil-based muds with comparable rheological properties performed about the same. Jalukar et al.[30] modified this model with a scaleup factor to correlate with the data obtained with the 8-in. TUDRP flow loop.

Zamora and Hanson,[31] on the basis of laboratory observations and field experience, compiled 28 rules of thumb to improve high-angle hole cleaning. Luo and Bern[32] presented charts to determine hole-cleaning requirements in deviated wells. These empirical charts were developed on the basis of the data collected with the BP 8-in. flow loop, and they predicted the critical flow rates required for prevention of cuttings-bed accumulation. The predictions have also been compared with some field data.

Mechanistic Modeling

The existing cuttings-transport correlations and/or models have a few empirical coefficients, determined based on laboratory and/or field data. There is a need for developing comprehensive cuttings transport mechanistic models that can be verified with experimental data. Different levels of the mechanistic approach are possible and can be built on gradually. Ideally, a fluid/solids interaction model, which would be coupled and integrated with a fluid-flow model to simulate the whole cuttings-transport process, is needed. Campos et al.[33] recently made such an attempt, but much more work is needed to develop a comprehensive solids/liquid flow model.

Ford et al.[34] published a model for the prediction of minimum transport velocity for two modes: cuttings suspension and cuttings rolling. The predictions were compared with laboratory data.

Gavignet and Sobey[35] presented a cuttings transport model based on physical phenomena, similar to that published by Wilson,[36] for slurry flow in pipelines that is known as the double-layer model. The model has many interrelated equations and a substantial number of parameters, a few of which are difficult to determine. Martin et al.[37] developed a numerical correlation based on the cuttings-transport data that they had collected in the laboratory and in the field.

Clark and Bickham[22] presented a cuttings-transport model based on fluid mechanics relationships, in which they assumed three cuttings-transport modes: settling, lifting, and rolling—each dominant within a certain range of wellbore angles. Predictions of the model were compared with critical and subcritical flow data they had collected with the TUDRP’s 5- and 8-in.flow loops. A prediction of the model was also used to examine several situations in which poor cuttings transport had been responsible for drilling problems.

Campos et al.[33] developed a mechanistic model for predicting the critical velocity as well as the cuttings-bed height for subcritical flow conditions. Their work was based on earlier work by Oraskar and Whitmore[38] for slurry transport in pipes. The model’s predictions are good for thin muds, but the model needs to be further refined to account for thick muds and pipe rotation.

Kenny et al.[39] defined a lift factor that they used as an indicator of cuttings-transport performance. The lift factor is a combination of the fluid velocity in the lower part of the annulus and the mud-settling velocity determined by Chien’s correlation.[20]

The cuttings-free mud flow creates a shear force at the interface with the moving cuttings bed, which drags the moving cuttings zone along with it. In the moving cuttings zone, gravity forces tend to make the cuttings fall onto the fixed cuttings bed, while aerodynamic and gel forces tend to keep the cuttings suspended. At the interface between the moving cuttings zone and the stationary cuttings bed, fluid friction is trying to strip off cuttings, which are held by gravity and cohesive forces. The balance of these forces determines whether the cuttings bed increases or decreases in depth. The critical flow rate for cuttings transport leaves the cuttings bed unchanged. For effective hole cleaning, the desired flow rate exceeds the critical flow rate.

Field Application

When the results of cuttings transport research and field experience are integrated into a drilling program, hole-cleaning problems are avoided, and excellent drilling performance follows. This has certainly been the case when engineers achieved two new world records in extended-reach drilling.

Guild and Hill[40] presented another example of integration of hole-cleaning research into field practice. They reported trouble-free drilling in two extended-reach wells after they lost one well because of poor hole cleaning. Their program was designed to maximize the footage drilled between wiper trips and eliminate hole-cleaning backreaming trips before reaching the casing point. They devised a creative way to avoid significant cuttings accumulation by carefully monitoring the pickup weight, rotating weight, and slackoff weight as drilling continued. They observed that cuttings accumulation in the hole caused the difference between the pickup weight and the slackoff weight to keep increasing, while cleaning the hole decreased the difference. By observing the changes in these parameters and by the use of other readily available information, they were able to closely monitor hole cleaning and control the situation.

Air, Mist, and Foam Drilling

Air and mist drilling have several advantages over conventional drilling fluids. The principle advantages are higher penetration rates, longer bit life, and no lost-circulation problems. The usual disadvantages are control of fluid influx and control of high-pressure zones.

To realize these advantages, it is important to maintain adequate circulation. Determining the required volume flow rate to maintain this "adequate" circulation has always been difficult. The best available technique has been the chart developed by R.R. Angel.[41] This chart allows the estimation of volume circulation rates for various hole sizes, drillpipe sizes, and penetration rates,

One difficulty with Angel’s result is that the equation giving the volume flow rate must be solved by trial and error. This difficulty is avoided by using the charts prepared by Angel, provided the case of interest is tabulated or can be estimated from similar cases. A second difficulty is that the drill cuttings are assumed to travel at the same velocity as the air. Angel notes that this is not a conservative assumption, and the analysis by Mitchell[42] demonstrates that the predicted flow rates are 20 to 30% too low. The downhole temperatures used for Angel’s chart are assumed to be 80°F at the surface, increasing 1°F per 100 ft of depth. There is no convenient way to convert to other temperatures. A final consideration is that the Angel charts do not apply to mist drilling. The addition of water to the air requires increases in both the volume flow rate and standpipe pressures to maintain the same penetration rate.

Compressible Flow

The flow of a compressible fluid can often produce results that seem to go counter to common sense. For instance, consider the steady flow of air in a constant area duct. As with all fluids, there is a pressure loss because of friction, and the pressure decreases continuously from the entrance of the duct to the exit. Unlike the flow of incompressible fluids, the fluid velocity increases from the entrance of the duct to the exit. How could friction make the fluid speed up?

Two facts account for this acceleration. First, the gas pressure is proportional to the density (as in the ideal gas law P = ρRT). As the pressure of the gas decreases, the density must decrease also. Second, because the mass flow through the duct is constant, the product of density and velocity is constant. Thus, as the density decreases with the pressure, the velocity must increase to maintain the mass flow.

This example demonstrates a typical compressible flow characteristic—the interrelationship of pressure and mass flow. In air drilling, high velocities are needed at bottomhole to remove the cuttings. High velocities result in friction-pressure drops in the drillpipe and annulus, so higher standpipe pressures may be needed to keep the air flowing. Higher standpipe pressures result in higher gas densities, and, hence, result in lower velocities. Fortunately, most air-drilling operations do not result in the vicious circle situation previously described.

Cuttings Transport and Mist Flow in Vertical Wells

The addition of the effect of cuttings and mist to the equations already developed require two changes. First, the effect of the cuttings and mist on momentum equation must be accounted for; and second, the forces exerted on the cuttings and mist must be determined. The principles of multiphase flow can be applied to both of these effects.

Two basic ideas are sufficient to develop the modified momentum equation. First, the mass flow rate of the cuttings is easy to determine; it is the product of the penetration rate, the hole area, and the density of the rock. Assuming that the cuttings velocity is known, a "density" for the cuttings mass flow rate can be determined.

....................(3.150)

This density represents the total mass of cuttings in a volume of the duct divided by the volume of the duct. The ratio of this density to the actual density of the rock is the volume fraction of the cuttings,

....................(3.151)

The remainder of the volume is filled by the air, with an air in-mixture density defined as

....................(3.152)

With these definitions, the cuttings transport equivalents to the single-phase flow equations can be written as

....................(3.153)

....................(3.154)

....................(3.155)

and

....................(3.156)

Note that G and Gs are constant. The final missing piece is the relationship between the velocity of the air and the velocity of the cuttings. There is a large body of literature on the data necessary to determine this relationship. For example, in the petroleum engineering literature, there is the work of Gray.[43] There is also a large amount of literature on terminal settling velocities for solid particles (see Bourgoyne[3], Chap. 1, Sec. 3, pages 172–174). The equation is given by Gray as Eq. 15 of the Appendix to Gray.[43] Rewritten in terms of flow variables previously defined, this equation becomes

....................(3.157)

where

....................(3.158)

and

....................(3.159)

The term Ws is the buoyant weight of the cuttings. The term P is the aerodynamic force exerted on the cuttings by the air, with CD the drag coefficient and δ the ratio of the average particle volume to its cross-sectional area. Values of CD can be found for various types of rock in Bourgoyne[3], pages 172–174. The term δ was evaluated for an average cutting diameter of 3/8 in. This size is considered to be typical of cuttings at the bit. Higher up the hole, these cuttings get broken into smaller pieces. Because there is no way of predicting the change in average particle size as the cuttings move up the annulus, the average diameter is held fixed at 3/8 in. This assumption causes the model to overpredict the relative velocity between the air and the cuttings. This assumption is conservative because higher air velocities are now needed to lift the cuttings. The assumption used by Angel[41] is that the particle velocity and the air velocity are equal, and he notes that this is not conservative.

The addition of mist to the flowing equations is much simpler than adding the cuttings. The water droplets in a mist are very small, and, as a result, the relative velocity between the air and the mist droplets is small. The usual assumption used in two-phase flow analysis is that the air and mist move at the same velocity, and simulations using Eq. 3.22 verify this. Eqs. 3.17 through 3.19 are suitable to model mist flow with the following changes: the mass flow and density of the mist replace those for the cuttings, and the velocities of the mist and the air are set equal.

This cuttings model predicts higher-volume flow rates than Angel’s model, which was expected because of the conservative nature of the cuttings model. The cuttings model also shows, however, that the flow rates specified by Angel are adequate to clean the bole, even though they do not satisfy the 3,000 ft/min requirement. The predicted temperatures are reasonably near the undisturbed geothermal temperature, which justifies the temperature assumptions used by Angel.

Foams

Foams are being used in a number of petroleum industry applications that exploit their high viscosity and low liquid content. Some of the earliest applications for foam dealt with its use as a displacing agent in porous media and as a drilling fluid. Following these early applications, foam was introduced as a wellbore circulating fluid for cleanout and workover applications. In the mid-1970s, N2-based foams became popular for both hydraulic fracturing and fracture-acidizing stimulation treatments. In the late 1970s and early 1980s, foamed cementing became a viable service, as did foamed gravel packing. Most recently, CO2 foams have been found to exhibit their usefulness in hydraulic fracturing stimulation.

Regardless of why they are applied, these compressible foams are structured, two-phase fluids that are formed when a large internal phase volume (typically 55 to 95%) is dispersed as small discrete entities through a continuous liquid phase. Under typical formation temperatures of 90°F (32.2°C) encountered in stimulation work, the internal phases N2 or CO2 exist as a gas and, hence, are properly termed foams in their end-use application. However, the formations of such fluids at typical surface conditions of 75°F (23.9°C) and 900 psi [6205 kPa] produce N2 as a gas but CO2 as a liquid. A liquid/liquid two-phase structured fluid is classically called an emulsion. The end-use application of the two-phase fluid, however, normally is above the critical temperature of CO2 at which only a gas can exist, so we consider the fluids together as foams. The liquid phase typically contains a surfactant and/or other stabilizers to minimize phase separation (or bubble coalescence).

These dispersions of an internal phase within a liquid can be treated as homogeneous fluids, provided bubble size is small in comparison to flow geometry dimensions. Volume percent of the internal phase within a foam is its quality. The degree of internal phase dispersion is its texture. At a fixed quality, foams are commonly referred to as either fine or coarse textured. Fine texture denotes a high level of dispersion characterized by many small bubbles with a narrow size distribution and a high specific surface area, and coarse texture denotes larger bubbles with a broad size distribution and a lower specific surface area.

Because foams exhibit shear-rate-dependent viscosities in laminar flow, they are classified as non-Newtonian fluids. In addition to shear rate, their apparent viscosities also appear to be dependent on quality, texture, and liquid-phase rheological properties. Measured laminar-flow apparent viscosities generally are larger than those of either constituent phase at all shear rates. When the liquid phase is thickened by addition of solids, soluble high-molecular-weight polymers, or other viscosifying agents, we see production of even larger foam viscosities. While laminar flow is characterized by strictly viscous energy dissipation, turbulent flow is characterized more by kinetic than viscous energy dissipation. Density and velocity are the factors that establish kinetic energy, and reduced foam density may outweigh an increased viscosity contribution and produce a turbulent-flow friction loss less than liquid-phase friction loss. Soluble high-molecular-weight polymers produce a form of turbulent drag reduction that is analogous to that which occurs in a nonfoamed liquid. In this case, a substantial drag-reduction effect is evident when one compares the turbulent-flow friction loss of foams with and without a gelled liquid phase.

Interactions between forces caused by surface tension, viscosity, inertia, and buoyancy produce a variety of effects observable in foams. These effects include different bubble shapes and sizes. Anomalous effects have been attributed to slippage as well as bubble size or texture. Buoyancy and inertia forces act on the foam and tend to destroy the discrete bubble structure, which makes the foam dynamically unstable. However, when work is performed on foam, as is the case when foam flows in a pipe, the bubble structure is being destroyed dynamically and then rebuilt, making the foam macroscopically act as a homogeneous fluid.

Beyer et al.[44] developed foam flow equations from data collected in horizontal pipes. They observed slippage, applied Mooney’s method for flow data correction, and correlated the data with a Bingham plastic flow model. Blauer et al.[45] concluded that foam behaves as a Bingham plastic without slippage in laminar flow. They equated the Buckingham-Reiner equation to the Hagen-Poiseuille equation to determine an expression for effective viscosity for use in conventional Newtonian fluid laminar and turbulent-flow friction-loss relationships.[8] A critical Reynold’s number of 2,100 was used to denote transition from laminar to turbulent flow. To the best of our knowledge, they present the only experimental turbulent foam flow data in the literature. Sanghani and Ikoku[46] studied the rheological properties of foam flowing in an annulus and concluded that foam rheological behavior was best represented as a pseudoplastic fluid. They also stated that their data could be represented by a Bingham plastic model and a yield pseudoplastic model without large errors.

Earlier investigators noted drag-reduction effects in turbulent two-phase flows when drag-reducing additives were introduced into the liquid phase. These investigators, however, did not deal directly with foam flow but with such diverse two-phase flow regimes as slug, plug, and annular mist.

The importance of foam rheological properties has been recognized by investigators; however, very little agreement exists among them. Foam has been characterized as a Bingham plastic, a pseudoplastic, and a yield pseudoplastic. Slippage has been observed in some, but not all, cases. Unexplained anomalous effects were observed in many cases. Bubble size and shape have been considered and neglected. All these vastly different observations indicate that foam is a very complex fluid that could exhibit a number of characteristics. All investigators agree, however, that a rheological dependence on quality exists. That foams in general exhibit a yield stress is also well supported. Investigations have been conducted primarily with water as the liquid phase.

Summary Guidelines for Efficient Hole Cleaning

Based on the results of many laboratories’ research and various field experiences and observations, the following general guidelines are recommended.

Design the well path so that it avoids critical angles, if possible.

Use top-drive rigs, if possible, to allow pipe rotation while tripping.

Use various hole-cleaning monitoring techniques including a drilled cuttings retrieval rate, a drilled cuttings physical appearance, pressure while drilling, and a comparison of pickup weight, slackoff weight, and rotating weight.

Perform wiper trips as the hole condition dictates.

Sample Calculations

Introduction

The most important consideration in making hydraulic calculations is the use of consistent units. Unfortunately, oilfield units are rarely consistent; in some cases they are unique to the industry. The universal set of consistent units is the SI Metric System of Units. The Society of Petroleum Engineers (SPE) has available a publication: "The SI Metric System of Units and SPE Metric Standard" that contains every conversion factor necessary. Whenever there is a question of units, the safest solution is to convert all units to SI units, solve the problem, and then convert the answer back to the common engineering units.

Sample Problem

Geometry. A deviated well kicks off at 3,000 ft and is drilled to total depth (TD) at an angle of 30° to the vertical. The well’s total measured depth is 11,000 ft. The well is cased with 72-ppf 13 3/8-in. casing (13.375 × 12.347 in.) set at 3,000 ft. The drillstring consists of 900 ft of 8-in. 147-ppf drill collars (8 × 3 in.), 19 ½-ppf drillpipe (5 × 4.206 in.), a 9 5/8 -in. bit with 3× 13/22-in. nozzles. The undisturbed temperature is 70°F at the surface with a 1.4°F/100-ft gradient. We will neglect the build section and assume the well trajectory is vertical to 3,000 ft measured depth, and deviated at 30° to the vertical from 3,000 ft measured depth to 11,000 ft measured depth. We will assume the open hole is gauge (9.625 in.).

2. Assume a layered wellbore with 14 lbm/gal mud from surface to 5,000 ft (measured depth) and 9 lbm/gal mud from 5,000 ft to TD. What is the pressure at TD?

For layer 1:

For layer 2:

3. Assume the wellbore is filled with nitrogen with a surface pressure of 2,000 psi. What is the pressure at TD? This problem is much more difficult because the gas density and temperature vary over the length of the wellbore. The pressure change is given by

Frictional Pressure Loss. 4. Assume fresh water is being circulated at 600 gal/min. What is the pressure change inside a single vertical 30-ft joint of drillpipe? Assume the density is 8.34 lbm/gal and the viscosity is 1 cp.

This Reynolds number indicates turbulent flow. To determine the friction factor, first determine the relative roughness k/D. From Fig. 3.10, the relative roughness is about .0004 for commercial steel. The friction factor is about .011 from Fig. 3.9. Friction pressure drop is given by

The hydrostatic pressure change per foot is

Total pressure change per length of pipe for flow downward is

The total pressure change in a 30-ft pipe joint is 0.166 × 30 = 4.98 psi.

5. Assume a 10-lbm/gal mud is being circulated at 100 gal/min. What is the frictional pressure change in the annulus outside a single 30-ft joint of drillpipe? Use the Bingham plastic model and assume the plastic viscosity is 40 cp and the YP is 15 lbf/100 ft 2.

6. Repeat Calculation 5, but assume the fluid is a power-law fluid. Remember that PV and YP were determined from the 300-rpm and 600-rpm readings of the Fann viscometer. The equivalent shear stresses are

7. For a flow rate of 600 gal/min, what is the fluid pressure in the bit nozzles? The mud density is 12 lbm/gal. What is the pressure recovery in the annulus?

Nomenclature

a

=

acoustic velocity, m/s

αvs , bvs

=

constants that include the viscometer dimensions, the spring constant, and all conversion factors

A

=

flow area (see subscripts), m2

c

=

average concentration of cuttings overall

ca

=

cuttings concentration in annular region

co

=

feed concentration of cuttings

cp

=

cuttings concentration in plug region

C

=

compressibility

Cd

=

discharge coefficients for the flow through an area change, dimensionless

Clark, R.K. and Fontenot, J.E. 1974. Field Measurements of the Effects of Drillstring Velocity, Pump Speed, and Lost Circulation Material on Downhole Pressures. Presented at the Fall Meeting of the Society of Petroleum Engineers of AIME, Houston, 6-9 October. SPE-4970-MS. http://dx.doi.org/10.2118/4970-MS.

David, A. and Marsden, S.S. Jr. 1969. The Rheology of Foam. Presented at the Fall Meeting of the Society of Petroleum Engineers of AIME, Denver, 28 September–1 October. SPE-2544-MS. http://dx.doi.org/10.2118/2544-MS.

Ravi, K. and Sutton, D.L. 1990. New Rheological Correlation for Cement Slurries as a Function of Temperature. Paper presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 23–26 September.

Reynolds, O. 1883. An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous, and the Laws of Resistance in Parallel Channels. Trans., Royal Society, London, 174.